Show that an algebraically closed field must be infinite.
Answer
If F is a finite field with elements $a_1, … , a_n$ the polynomial $f(X)=1 + \prod_{i=1}^n (X – a_i)$ has no root in F, so F cannot be algebraically closed.
My Question
Could we not use the same argument if F was countably infinite? Couldn't we say that if F was a field with elements $a_1, a_2, … $ then the polynomial $f(X) = 1 + \prod_{i=1}^{\infty} (X – a_i)$ does not split over F?
Thank you in advance
Best Answer
We can't, because $\prod_{i=1}^\infty (X-a_i)$ is not a polynomial. Infinite combinations of finitary operations are not defined; when we do talk about them, what is really going on is that we are taking some sort of limit, but to do that we need topology to be present, and even then, the infinitary operation will be subtle. A priori, one cannot talk about an infinite sum or an infinite product of elements.